The rotating angle suggests immediately the domain and definition of Sin(x), Cos(x) using simple projections. And, huuuuge bonus, that's a right triangle.

I just think lesson zero in the above article should have been these projections, and a simple "From this all angles can be defined as right triangles in your preferred coordinate system", and you're off to the races in any old direction.

This is not how I was taught it, and I'm retrospectively upset.

> Instead of adding two separate angles α and β, we’ll use θ and −θ.

It's not clear that the geometric proof based on pictures with positive angles α and β also applies to a negative β. I think one should at least provide a separate picture of that case...

> If you allow negative side lengths, as you should [...]

I mean, the blog post was about a pedagogically better suited construction of trig functions. If we allow negative side lengths, it's not really going to be intuitive anymore. At this point, you might just as well use the power series definition (or the one based on the complex exp function) since it makes things easy to prove.

Imaginary lengths may be unintuitive but negative is pretty simple.

The idea of negative measurements (meaning "opposite direction along a line") as is well understood before children study geometry. Kindergarteners learn "left" vs "right" and "forward" vs "backward".

I agree with other poster about the value of showing negative and positive lengths visually in the same diagram.

Magnitudes are intuitionally directionless (hence why norms and measures are nonnegative reals). Of course, you can extend notions suitably and make it work, but I really don't think that this is terribly intuitive. (Of course, different people will find different things intuitive.)